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Theorem isstruct2 16269
 Description: The property of being a structure with components in (1st ‘𝑋)...(2nd ‘𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.)
Assertion
Ref Expression
isstruct2 (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))

Proof of Theorem isstruct2
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brstruct 16268 . . 3 Rel Struct
21brrelex12i 5407 . 2 (𝐹 Struct 𝑋 → (𝐹 ∈ V ∧ 𝑋 ∈ V))
3 ssun1 3999 . . . . 5 𝐹 ⊆ (𝐹 ∪ {∅})
4 undif1 4267 . . . . 5 ((𝐹 ∖ {∅}) ∪ {∅}) = (𝐹 ∪ {∅})
53, 4sseqtr4i 3857 . . . 4 𝐹 ⊆ ((𝐹 ∖ {∅}) ∪ {∅})
6 simp2 1128 . . . . . . 7 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → Fun (𝐹 ∖ {∅}))
76funfnd 6168 . . . . . 6 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (𝐹 ∖ {∅}) Fn dom (𝐹 ∖ {∅}))
8 elinel2 4023 . . . . . . . . . . 11 (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → 𝑋 ∈ (ℕ × ℕ))
9 1st2nd2 7486 . . . . . . . . . . 11 (𝑋 ∈ (ℕ × ℕ) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
108, 9syl 17 . . . . . . . . . 10 (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
11103ad2ant1 1124 . . . . . . . . 9 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
1211fveq2d 6452 . . . . . . . 8 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (...‘𝑋) = (...‘⟨(1st𝑋), (2nd𝑋)⟩))
13 df-ov 6927 . . . . . . . . 9 ((1st𝑋)...(2nd𝑋)) = (...‘⟨(1st𝑋), (2nd𝑋)⟩)
14 fzfi 13094 . . . . . . . . 9 ((1st𝑋)...(2nd𝑋)) ∈ Fin
1513, 14eqeltrri 2856 . . . . . . . 8 (...‘⟨(1st𝑋), (2nd𝑋)⟩) ∈ Fin
1612, 15syl6eqel 2867 . . . . . . 7 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (...‘𝑋) ∈ Fin)
17 difss 3960 . . . . . . . . 9 (𝐹 ∖ {∅}) ⊆ 𝐹
18 dmss 5570 . . . . . . . . 9 ((𝐹 ∖ {∅}) ⊆ 𝐹 → dom (𝐹 ∖ {∅}) ⊆ dom 𝐹)
1917, 18ax-mp 5 . . . . . . . 8 dom (𝐹 ∖ {∅}) ⊆ dom 𝐹
20 simp3 1129 . . . . . . . 8 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → dom 𝐹 ⊆ (...‘𝑋))
2119, 20syl5ss 3832 . . . . . . 7 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → dom (𝐹 ∖ {∅}) ⊆ (...‘𝑋))
2216, 21ssfid 8473 . . . . . 6 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → dom (𝐹 ∖ {∅}) ∈ Fin)
23 fnfi 8528 . . . . . 6 (((𝐹 ∖ {∅}) Fn dom (𝐹 ∖ {∅}) ∧ dom (𝐹 ∖ {∅}) ∈ Fin) → (𝐹 ∖ {∅}) ∈ Fin)
247, 22, 23syl2anc 579 . . . . 5 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (𝐹 ∖ {∅}) ∈ Fin)
25 p0ex 5097 . . . . 5 {∅} ∈ V
26 unexg 7238 . . . . 5 (((𝐹 ∖ {∅}) ∈ Fin ∧ {∅} ∈ V) → ((𝐹 ∖ {∅}) ∪ {∅}) ∈ V)
2724, 25, 26sylancl 580 . . . 4 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → ((𝐹 ∖ {∅}) ∪ {∅}) ∈ V)
28 ssexg 5043 . . . 4 ((𝐹 ⊆ ((𝐹 ∖ {∅}) ∪ {∅}) ∧ ((𝐹 ∖ {∅}) ∪ {∅}) ∈ V) → 𝐹 ∈ V)
295, 27, 28sylancr 581 . . 3 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → 𝐹 ∈ V)
30 elex 3414 . . . 4 (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → 𝑋 ∈ V)
31303ad2ant1 1124 . . 3 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → 𝑋 ∈ V)
3229, 31jca 507 . 2 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (𝐹 ∈ V ∧ 𝑋 ∈ V))
33 simpr 479 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → 𝑥 = 𝑋)
3433eleq1d 2844 . . . 4 ((𝑓 = 𝐹𝑥 = 𝑋) → (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ↔ 𝑋 ∈ ( ≤ ∩ (ℕ × ℕ))))
35 simpl 476 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → 𝑓 = 𝐹)
3635difeq1d 3950 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → (𝑓 ∖ {∅}) = (𝐹 ∖ {∅}))
3736funeqd 6159 . . . 4 ((𝑓 = 𝐹𝑥 = 𝑋) → (Fun (𝑓 ∖ {∅}) ↔ Fun (𝐹 ∖ {∅})))
3835dmeqd 5573 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → dom 𝑓 = dom 𝐹)
3933fveq2d 6452 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → (...‘𝑥) = (...‘𝑋))
4038, 39sseq12d 3853 . . . 4 ((𝑓 = 𝐹𝑥 = 𝑋) → (dom 𝑓 ⊆ (...‘𝑥) ↔ dom 𝐹 ⊆ (...‘𝑋)))
4134, 37, 403anbi123d 1509 . . 3 ((𝑓 = 𝐹𝑥 = 𝑋) → ((𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥)) ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
42 df-struct 16261 . . 3 Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
4341, 42brabga 5228 . 2 ((𝐹 ∈ V ∧ 𝑋 ∈ V) → (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
442, 32, 43pm5.21nii 370 1 (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  Vcvv 3398   ∖ cdif 3789   ∪ cun 3790   ∩ cin 3791   ⊆ wss 3792  ∅c0 4141  {csn 4398  ⟨cop 4404   class class class wbr 4888   × cxp 5355  dom cdm 5357  Fun wfun 6131   Fn wfn 6132  ‘cfv 6137  (class class class)co 6924  1st c1st 7445  2nd c2nd 7446  Fincfn 8243   ≤ cle 10414  ℕcn 11378  ...cfz 12647   Struct cstr 16255 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-er 8028  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-nn 11379  df-n0 11647  df-z 11733  df-uz 11997  df-fz 12648  df-struct 16261 This theorem is referenced by:  structn0fun  16271  isstruct  16272  setsstruct2  16297
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